Chapter 8

Suppose that the equation \(p(x)=-2 x^{2}+280 x-1000\), where \(x\) represents the number of items sold, describes the profit function for a certain business. How many items should be sold to maximize the profit?

If \(f(x)=3 x^{2}-2 x-1\) and \(g(x)=x\), find \(f \circ g\) and \(g \circ f\). (Recall that we have previously named \(g(x)=x\) the "identity function.")

Suppose that the cost function for the production of a particular item is given by the equation \(C(x)=2 x^{2}-\) \(320 x+12,920\), where \(x\) represents the number of items. How many items should be produced to minimize the cost?

(a) Graph \(f(x)=|x|, f(x)=2|x|, f(x)=4|x|\), and \(f(x)=\frac{1}{2}|x|\) on the same set of axes. (b) Graph \(f(x)=|x|, f(x)=-|x|, f(x)=-3|x|\), and \(f(x)=-\frac{1}{2}|x|\) on the same set of axes. (c) Use your results from parts (a) and (b) to make a conjecture about the graphs of \(f(x)=a|x|\), where \(a\) is a nonzero real number. (d) Graph \(f(x)=|x|, f(x)=|x|+3, f(x)=|x|-4\), and \(f(x)=|x|+1\) on the same set of axes. Make a conjecture about the graphs of \(f(x)=|x|+k\), where \(k\) is a nonzero real number. (e) Graph \(f(x)=|x|, f(x)=|x-3|, f(x)=|x-1|\), and \(f(x)=|x+4|\) on the same set of axes. Make a conjecture about the graphs of \(f(x)=|x-h|\), where \(h\) is a nonzero real number. (f) On the basis of your results from parts (a) through (e), sketch each of the following graphs. Then use a graphing calculator to check your sketches. (1) \(f(x)=|x-2|+3\) (2) \(f(x)=|x+1|-4\) (3) \(f(x)=2|x-4|-1\) (4) \(f(x)=-3|x+2|+4\) (5) \(f(x)=-\frac{1}{2}|x-3|-2\)

For each of the following, predict the general shape and location of the graph, and then use your calculator to graph the function to check your prediction. (Your knowledge of the graphs of the basic functions that are being added or subtracted should be helpful when you are making your predictions.) (a) \(f(x)=x^{4}+x^{2}\) (b) \(f(x)=x^{3}+x^{2}\) (c) \(f(x)=x^{4}-x^{2}\) (d) \(f(x)=x^{2}-x^{4}\) (e) \(f(x)=x^{2}-x^{3}\) (f) \(f(x)=x^{3}-x^{2}\) (g) \(f(x)=|x|+\sqrt{x}\) (h) \(f(x)=|x|-\sqrt{x}\)

4 Neglecting air resistance, the height of a projectile fired vertically into the air at an initial velocity of 96 feet per second is a function of time \(x\) and is given by the equation \(f(x)=96 x-16 x^{2}\). Find the highest point reached by the projectile.

For each of the following, find the graph of \(y=\) \((f \circ g)(x)\) and of \(y=(g \circ f)(x)\). (a) \(f(x)=x^{2}\) and \(g(x)=x+5\) (b) \(f(x)=x^{3}\) and \(g(x)=x+3\) (c) \(f(x)=x-6\) and \(g(x)=-x^{3}\) (d) \(f(x)=x^{2}-4\) and \(g(x)=\sqrt{x}\) (e) \(f(x)=\sqrt{x}\) and \(g(x)=x^{2}+4\) (f) \(f(x)=\sqrt[3]{x}\) and \(g(x)=x^{3}-5\)

Find two numbers whose sum is 30 , such that the sum of the square of one number plus ten times the other number is a minimum.

Find two numbers whose sum is 50 and whose product is a maximum.

48\. Find two numbers whose difference is 40 and whose product is a minimum

49\. Two hundred and forty meters of fencing is available to enclose a rectangular playground. What should be the dimensions of the playground to maximize the area?

Motel managers advertise that they will provide dinner, dancing, and drinks for \(\$ 50\) per couple for a New Year's Eve party. They must have a guarantee of 30 couples. Furthermore, they will agree that for each couple in excess of 30 , they will reduce the price per couple by \(\$ 0.50\) for all attending. How many couples will it take to maximize the motel's revenue?

\(f(x)=\frac{2 x}{(x-2)(x+3)}\)

A cable TV company has 1000 subscribers, each of whom pays \(\$ 15\) per month. On the basis of a survey, the company believes that for each decrease of \(\$ 0.25\) in the monthly rate, it could obtain 20 additional subscribers. At what rate will the maximum revenue be obtained, and how many subscribers will there be at that rate? 1100 subscribers at \(\$ 13.75\) per month

A manufacturer finds that for the first 500 units of its product that are produced and sold, the profit is \(\$ 50\) per unit. The profit on each of the units beyond 500 is decreased by \(\$ 0.10\) times the number of additional units sold. What level of output will maximize profit?

Suppose your friend was absent the day this section was discussed. How would you explain to her the ideas pertaining to \(x\) intercepts of the graph of a function, zeros of the function, and solutions of the equation \(f(x)=0\) ?

Give a step-by-step explanation of how to find the \(x\) intercepts of the graph of the function \(f(x)=2 x^{2}+7 x-4\).

Give a step-by-step explanation of how to find the vertex of the parabola determined by the equation \(f(x)=\) \(-x^{2}-6 x-5\)

\(f(x)=\frac{4 x}{x^{2}-x-12}\)

Suppose that the viewing window on your graphing calculator is set so that \(-15 \leq x \leq 15\) and \(-10 \leq y \leq 10\). Now try to graph the function \(f(x)=x^{2}-8 x+28\). Nothing appears on the screen, so the parabola must be outside the viewing window. We could arbitrarily expand the window until the parabola appeared. However, let's be a little more systematic and use \(\left(-\frac{b}{2 a}, f\left(-\frac{b}{2 a}\right)\right)\) to find the vertex. We find the vertex is at \((4,12)\), so let's change the \(y\) values of the window so that \(0 \leq y \leq 25\). Now we get a good picture of the parabola. Graph each of the following parabolas, and keep in mind that you may need to change the dimensions of the viewing window to obtain a good picture. (a) \(f(x)=x^{2}-2 x+12\) (b) \(f(x)=-x^{2}-4 x-16\) (c) \(f(x)=x^{2}+12 x+44\) (d) \(f(x)=x^{2}-30 x+229\) (e) \(f(x)=-2 x^{2}+8 x-19\)

For each of the following quadratic functions, use the discriminant to determine the number of real-number zeros, and then graph the function with a graphing calculator to check your answer. (a) \(f(x)=3 x^{2}-15 x-42\) (b) \(f(x)=2 x^{2}-36 x+162\) (c) \(f(x)=-4 x^{2}-48 x-144\) (d) \(f(x)=2 x^{2}+2 x+5\) (e) \(f(x)=4 x^{2}-4 x-120\) (f) \(f(x)=5 x^{2}-x+4\)

\(f(x)=\sqrt{x^{2}+1}-4\)

\(f(x)=\sqrt{16-x^{2}} \quad[-4,4]\)

\(f(x)=\sqrt{1-x^{2}} \quad[-1,1]\)

Suppose that the profit function for selling \(n\) items is given by $$ P(n)=-n^{2}+500 n-61,500 $$ Evaluate \(P(200), P(230), P(250)\), and \(P(260)\). -1500; 600; 1000; 900

. The equation \(A(r)=\pi r^{2}\) expresses the area of a circular region as a function of the length of a radius \((r)\). Compute \(A(2), A(3), A(12)\), and \(A(17)\) and express your answers to the nearest hundredth. \(12.57 ; 28.27 ; 452.39 ; 907.92\)

The height of a projectile fired vertically into the air (neglecting air resistance) at an initial velocity of 64 feet per second is a function of the time \((t)\) and is given by the equation \(h(t)=64 t-16 t^{2}\). Compute \(h(1), h(2), h(3)\), and \(h(4) . \quad 48 ; 64 ; 48 ; 0\)

A car rental agency charges \(\$ 50\) per day plus \(\$ 0.32\) a mile. Therefore the daily charge for renting a car is a function of the number of miles traveled \((m)\) and can be expressed as \(C(m)=50+0.32 m\). Compute \(C(75)\), \(C(150), C(225)\), and \(C(650) . \quad \$ 74 ; \$ 98 ; \$ 122 ; \$ 258\)

The equation \(I(r)=500 r\) expresses the amount of simple interest earned by an investment of \(\$ 500\) for 1 year as a function of the rate of interest ( \(r\) ). Compute \(I(0.11), I(0.12), I(0.135)\), and \(I(0.15)\). \(\$ 55 ; \$ 60 ; \$ 67.50 ; \$ 75\)

Suppose the height of a semielliptical archway is given by the function \(h(x)=\sqrt{64-4 x^{2}}\), where \(x\) is the distance from the center line of the arch. Compute \(h(0), h(2)\), and \(h(4) . \quad 8 ; 4 \sqrt{3} ; 0\)

The equation \(A(r)=2 \pi r^{2}+16 \pi r\) expresses the total surface area of a right circular cylinder of height 8 centimeters as a function of the length of a radius \((r)\). Compute \(A(2), A(4)\), and \(A(8)\) and express your answers to the nearest hundredth. \(125.66 ; 301.59 ; 804.25\)

What does it mean to say that the domain of a function may be restricted if the function represents a real-world situation? Give three examples of such functions.

Does \(f(a+b)=f(a)+f(b)\) for all functions? Defend your answer.